Example of two converging series where partial sums change orders infinitely often

Question

Looking for example of two series where their partial sums swicth order iniftely often, i.e. let $_1s_k$ be the partial sum for the first series and like $_2s_k$ for the second series, then for numbers $k_1>k_2>k_3>k_4>k_5>\cdots k_n>\cdots$, we get alternating ordering of partial sums

$ _1{s}_{k_1} < _2{s}_{k_1}$

$ _1{s}_{k_2} > _2{s}_{k_2}$

$ _1{s}_{k_3} < _2{s}_{k_3}$

$ _1{s}_{k_4} > _2{s}_{k_4}$

$ _1{s}_{k_5} < _2{s}_{k_5}$

I have tried using $2^{-k}$ and $3^{-k}$ series, forming two new series by taking a term from each of $2^{-k}$ and $3^{-k}$ alternatingly, or some number of terms from each series alternatingly.

Answer 1

Consider the series $\sum_{k=0}^\infty a_k$ and $\sum_{k=0}^\infty b_k$ with $$a_0=\log 2,\quad a_k=0\ (k\geq1);\qquad b_k={(-1)^k\over k+1}\quad(k\geq0)\ .$$